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NPS-54-90-006 



NAVAL POSTGRADUATE SCHOOL 

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THE EFFECTS OF DIFFERENT PRODUCTION 

RATE MEASURES AND COST STRUCTURES 

ON RATE ADJUSTMENT MODELS 

Dan C. Boger 

and 

Shu S. Liao 



February 1990 



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Naval Air Systems Command 
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The Effects of Different Production Rate Measures and Cost Structures on Rate Adjustment Models 



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Dan C. Boger and Shu S. Liao 



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Production rate, rate adjustment, learning curve, cost estimating 



9. ABSTRACT (Continue on reverse it necessary and 'Identify by block number) 

The effect of production rate on the cost of weapon systems has attracted much attention in the cost estimating 
community in recent years. A variety of adjustments to weapon systems cost models have been proposed to reflect 
the impact of different production rates. The most popular solution is to add a rate term to the traditional learning 
curve model. This paper examines the effects of different rate measures and cost structures on rate adjustment 
models. Numerical examples illustrate that the production rate term should be measure as a ratio and not as an 
absolute quantity of a production lot or a period. The paper also points out that a rate adjustment model is 
appropriate only with data from plants which have not undergone changes in cost structure. 



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THE EFFECTS OF DIFFERENT PRODUCTION RATE MEASURES 
AND COST STRUCTURES ON RATE ADJUSTMENT MODELS 



Dan C. Boger 

and 
Shu S. Liao 



Department of Administrative Sciences 
Naval Postgraduate School 
Monterey, California 93943 



February 1990 



THE EFFECTS OF DIFFERENT PRODUCTION RATE MEASURES 
AND COST STRUCTURES ON RATE ADJUSTMENT MODELS 



ABSTRACT 

The effect of production rate on the cost of weapon systems has attracted much 
attention in the cost estimating community in recent years. A variety of adjustments to weapon 
systems cost models have been proposed to reflect the impact of different production rates. 
The most popular solution is to add a rate term to the traditional learning curve model. This 
paper examines the effects of different rate measures and cost structures on rate adjustment 
models. Numerical examples illustrate that the production rate term should be measured as 
a ratio and not as an absolute quantity of a production lot or a period. The paper also points 
out that a rate adjustment model is appropriate only with data collected from plants which have 
not undergone changes in cost structure. 



THE EFFECTS OF DIFFERENT PRODUCTION RATE MEASURES 
AND COST STRUCTURES ON RATE ADJUSTMENT MODELS 

The effect of production rate on the cost of weapon systems has attracted much 
attention in the cost estimating community in recent years. A variety of adjustments to weapon 
systems cost models have been proposed to reflect the impact of different production rates. 
The most popular solution is to add a rate term to the traditional learning curve model. The 
resulting learning curve model augmented with the production rate variable is usually referred 
to as a rate adjustment model. The purpose of this paper is to examine the theoretical 
underpinning of the production rate effect on weapon system cost and illustrate that the 
popular solution to the rate problem may result in erroneous conclusions. Numerical examples 
will be used to illustrate the potential problems of the popular approach to production rate 
adjustment. The paper concludes with a discussion of the scenarios in which the rate 
adjustment models may be utilized. 

CONCEPTUAL FOUNDATION OF PRODUCTION RATE EFFECT 

The conceptual foundation of the production rate impact on cost is related to 
economies of scale. In many industries that effect is well understood. High production rates 
allow greater use of facilities and greater specialization of labor. The increased volume of 
materials purchased reduces their unit cost. The increased volume of production activities 
spreads fixed overhead costs over a larger quantity of products produced. Taken together. 
all these effects work to increase efficiency and lower production costs (Bemis, 1981; Large, 
et al., 1974; Under and Wilbourn, 1973). 

It should be noted, however, that a plant with a higher production rate does not 



necessarily produce at a lower unit cost when compared to another plant. This point is 
illustrated in Figure 1 . Assume there are three plants capable of producing the same item, 
such as a missile. The Average Unit Cost curve for each plant is shown as AUC1, AUC2, and 
AUC3, respectively. If the output quantity were fixed at 25 units, then Plant 1 is the most 
efficient of the three plants. However, if the output level were fixed at the rate of 40 units per 
period, Plant 1 's unit cost would be higher than that of Plant 2, which is the most efficient of 
the three at that production quantity. This is consistent with economic theory, which says that, 
in general, there are both economies and diseconomies of scale. This phenomenon is 
recognized by the above analysts and is reflected in their use of this familiar U-shaped average 
cost curve to incorporate the effect of production rate into weapon systems cost models. 



Figure 1 
Average Unit Cost and Production Capacity 



Unit Cost 




10 



30 40 50 60 70 

Output Quantity 



80 



90 



The same theory of economies and diseconomies of scale is applicable to a single 
plant's expansion when it is operating beyond its efficient capacity level. This scenario has 



significant implications in weapon systems cost estimation. Recent experience has shown that 
production rates of major weapon systems are subject to continual adjustment, sometimes 
significantly. At the low end of the spectrum is the initial production rate. This is usually a 
function of early procurement funding constraints and the technical risk of building substantial 
numbers of newly developed Items before the design has fully matured. Thus low rate initial 
production avoids the risk of incurring costly retrofits to early production units. During this 
early stage of production, the amount of fixed costs may vary from period to period because 
of the changing production setup. At the upper boundary is the limitation of available plant 
capacity and the requirement for additional investments in tooling and facilities for capacity 
expansion. Additional investments in tooling and facilities alter the cost structure of the plant. 
The unit cost curve of a plant expanding its investment in tooling and facilities is equivalent to 
changing from AUC1 to AUC2 as shown in Figure 1. 

REVIEW OF RATE ADJUSTMENT MODELS 

Although studies of the effect of production rate change on weapon systems cost 
began as early as the 1950s (Hirsch, 1952; Alchian, 1963), and various models had been 
proposed, the most widely used rate adjustment model in use today was developed by 
augmenting the traditional learning curve model with a production rate term: 

Z = aX b R c = YR C (1) 

where, 

Z = unit cost of the Item with production rate as well as learning considered, 

X = cumulative quantity produced, 

R = production rate measure, 



Y = unit cost of the item with only learning considered, 

a = a constant, usually called the theoretical first unit cost, 

b = a parameter, usually called the slope of the learning curve, 

c = a parameter, usually called the slope of the production rate curve. 

Empirical work on this production rate/learning model was first conducted at RAND, but 
the model was later popularized by Bemis (1981). Large, et al. (1974) attempted to develop 
this model for various production cost elements. They were forced to conclude, however, that 
the production-rate/cost relationship could not be predicted with any reasonable degree of 
confidence. For production planning purposes, they recommended that production rate effects 
in aircraft production programs be ignored because they were dominated by other effects. 
They also suggested that production rate is subject to change and, hence, is difficult to predict. 

Further work on the production rate/learning model was carried out by Smith (1976). 
He analyzed three aircraft programs for which a large number of data values were available 
due to long production periods. Where the data permitted, Smith applied his model separately 
to fabrication and assembly labor hours. He then compared his production rate/learning model 
to a reduced, learning-only model. Smith found that the rate term was an important contributor 
to the explanatory power of the model. However, he obtained a surprisingly large variation in 
parameter values for cases with similar production quantities and rates. Additional efforts using 
this approach were carried out by Bemis (1981), Cox and Gansler (1981), and others. 

If one recognizes the inherent rate instability scenario of major weapon systems 
production and the resultant changing cost structure discussed in the preceding section, then 
none of the inconclusive findings discussed above would be surprising. In the following 
sections, we will examine the issues of alternative production rate measures and changing cost 



structures, and we will discuss other major considerations that must be addressed before one 
can use the rate adjustment model in weapon systems cost estimation. 

ALTERNATIVE PRODUCTION RATE MEASURES 

Although the concept of production rate is clear, its measurement is by no means 
unambiguous. Several alternatives have been used as surrogate measures of production rate. 
The two primary measures are lot size and annual/monthly production quantity. We will first 
discuss these two and related measures, along with the difficulties of their use. We then 
discuss a third alternative, a ratio measure which we believe will avoid some of the difficulties 
of the measures used to date. 

Using Lot Size or Annual/Monthly Quantity As the Rate Measure 

Hirsch (1952), Cox and Gansler (1981), and Bohn and Kratz (1984) all used lot size as 
their measure of production rate. Hirsch was careful to note that his lot intervals were fairly 
stable; however, this has not been the case with almost all more-recent aircraft programs. 
Since the time (and, hence, cost) required to produce sequential, similarly-sized lots often 
changes over the life of the program, it is unclear what is being measured by the lot size 
proxy. 

Perhaps the most common measure of rate is that of production quantity in some time 
interval. The time period involved is usually selected as a function of data availability. Most 
studies use annual quantities as a measure of production rates. An inverse of the quantity- 
per-unit-time measure has also been used; Large, et ai. (1974) used the number of months 
required to reach a certain cumulative production quantity as their inverse measure. Some 



studies, such as Womer (1984), use monthly data. Womer notes that if there is substantial 
work-in-progress and the production period is long compared to the period of observation, then 
units produced in the following time period actually reflect work performed in the preceding 
time period, and this can result in substantial bias in estimation. Since this problem is 
especially critical for monthly data, Womer used a lagged model of production to obtain his 
estimates. 

When analyzing a cross-section of programs, it is possible to use an average rate for 
each program. Because the production rate may change in a typical production run, an 
average rate for an individual program is usually used in these cross-section analyses. Use 
of an average may understate the effects of these disruptive rate changes, but we do not 
expect it to mask the effect of production rate itself. Large, et al. (1974) used this approach 
in their examination of several programs. 

Gulledge and Womer (1986) noted that cumulative quantity is highly correlated with any 
of the production rate measures discussed above. Hence, using either the lot size or 
monthly/annual quantities as the measure of R in Equation (1) will produce unreliable models 
due to this collinearity of the cumulative quantity measure of learning (X) and the measure of 
production rate (R). The presence of this collinearity has resulted in the inability of analysts 
to separate statistically the effects of learning and production rate. For example, Large et al. 
(1974) concluded that the influence of production rate could not be estimated with confidence. 

Using a Ratio as the Rate Measure 

An alternative to the above measures which will tend to mitigate the multicollinearity 
problem is that of a ratio of the above production rate measures. This use of a ratio, if keyed 



to a base production rate, as the rate curve measure appears to be an innovation in the 
literature. Bemis (1981) uses the ratio of new rate to present rate as the rate measure, which 
is more a measure of rate change than a measure of the rate per se. A similar measure was 
adopted by Balut (1981) and Balut, et al. (1989); they used a ratio of old-to-new lot sizes to 
account for rate effects in an aircraft repricing model which also included a learning curve. 
On the other hand, Boger and Liao (1988) proposed using a standard, base, or predetermined 
rate as the denominator in the ratio and either lot sizes or annual/monthly quantities as the 
numerator. The advantage of using a base rate is that if one uses the rate to which the 
manufacturer has tooled the production facility as the base rate, then ratios greater than unity 
would indicate decreasing returns to variable inputs and ratios lesser than unity would indicate 
increasing returns to variable inputs. 

In addition to the mitigation of statistical problems, the use of a ratio as the rate 
measure has some intuitive advantages for cost estimating purposes. While the general 
formulation shown in Equation (1) for production rate is widely used, little has been done to 
examine the empirical implications of adding the production rate factor to the well known 
learning curve model. The definition of the parameter a of Equation (1) (referred to as the 
theoretical first unit cost in learning curve theory) is the unit cost when X = 1 and R = 1. While 
this interpretation seems logical, it does result in some awkward numbers because R = 1 is not 
close to the relevant production range for most of the production rate measures used in 
practice. It is, however, for our proposed measure. This issue can be illustrated with a simple 
example. This example will use a minimum of data points since this is the typical situation 
faced by cost analysts. 



An Illustrative Example 

Assume that the data for the first two production contracts for a new weapon system 
are as follows: 

Lot # Quantity Unit Price Algebraic Lot Midpoint 

1 100 $43,773 33.9 

2 100 31,035 147.0 

The algebraic lot midpoint is that quantity on the learning curve which corresponds to the 

average cost for that entire lot. Liao (1988 and 1989) provides detailed discussions of this 

concept and its measurement. 

A. Ratio Rate Measure -- Since there are only two data points, only the learning curve 

slope may be estimated at this point. We may use the following formula to determine the 

learning curve slope: 

Log (Y 2 / Y,) 
b = - (2) 

Log (M 2 / M,) 

where Y; and Mj represent the unit price and the algebraic midpoint of each lot respectively. 

The slope of the learning curve for our illustrative data may now be determined as follows: 

31 ,035 

L °9 43-773 

b = -:-%--- = -0.234422 or 85% curve 

. 147.0 

Log -33-g 

The first unit cost can be readily obtained by substituting the value of b into the basic learning 
curve equation: 

43,773 = a (33.9)-° 234422 

a = 100,000 



Note that implicit in the above computation is the production rate of 100 units. In other words, 
the $100,000 represents the cost of producing the first unit when the rate is 100 units per 
year. 

Let us assume that for year 3 requirements the government solicits step-ladder quotes 
from a potential contractor for this system. Step-ladder quotes are the quotes in a schedule 
of bids from a potential contractor for varying percentages of the government's planned total 
requirement for that year. (A full set of quotes, using a 10% step, would give the potential 
contractor's prices for 10%, 20%,. . ., and 100% of the government's requirement.) The 
differences in the prices quoted by a single contractor for various quantity levels during this 
single year, in principle, should reflect only the production rate effect. Let us further assume 
that the slope for the rate curve is 80%. If we want to evaluate the reasonableness of quotes 
at different production rate levels, the most logical approach is to anchor the rate measure at 
a given level within the relevant rate range, e.g., 100 units (base rate = 100), and measure 
different quantity levels as a ratio of that base rate. If the rate curve is known or agreed upon 
by both parties, the reasonable quotes for various quantity levels may be directly calculated 
by using the following formula: 

Z = Yr d (3) 

where, 

r = the slope of the production rate curve, and 

d = the logarithm of R (the ratio measure of rate) divided by the logarithm of 2. 
For example, with the assumed 80% rate curve, 85% learning curve, and a= 100,000, the 
reasonable quote for 300 units may be computed as follows: 

Zaoo = 25,554(0.8) IOfl(3)/,o « (2) = 25,554(0.8) 1585 = 17,942 



If the parameter value of the rate term is unknown, it can be estimated from annual step 
ladder quotes as follows. Since we define Z = aX b R c or YR C , the ratio of reasonable bid prices 
at various quantity levels as a function of the long-term learning curve may be determined as 
follows: 

R c = Z/Y, or Z/aX b (4) 

We may use the computed ratios for various quantity levels to determine the parameter value 
for the rate term. Table 1 shows the procedures described above. 









Table 1 








Estimating 


Rate Effect from Year 3 Step-Ladder Quotes 






(a = 100,000, 


Total Previous Quantity = 


200 units) 




Quote 






aX b 


aX b R c 


R c 


R 


Quantity 


Midpoint 




(Y) 


(Z) 


Z/Y Q/1 00 


50 


224.9 




28,088 


35,111 


1.250 


0.5 


100 


248.4 




27,442 


27,442 


1.000 


1 


200 


293.5 




26,390 


21,112 


0.800 


2 


300 


336.7 




25,554 


17,942 


0.702 


3 


400 


378.6 




24,861 


15,911 


0.640 


4 


500 


419.5 




24,269 


14,456 


0.576 


5 



Figure 2 shows the relationship between Z/Y and the rate measure, R. Note that the 
reasonable quotes should reflect a straight line on a log-log graph as shown in Figure 2. The 
slope of the rate curve can be derived from the values of the last two columns of Table 1 in 
the same way that the learning curve slope is usually derived (by using the log-linear 
regression method). In our case, the regression yields the exponent, c, -0.3218, which 
represents an 80% curve, the slope we used to generate the hypothetical data. 

B. Absolute Size Rate Measure - If we use the lot size or annual/monthly quantity 



10 



Figure 2 
Production Rate Curve (80%) 



10 



0.1 







i i i i i i"T— »4j 


Base ratje (100 uinitjs): 

7 : 






- ] | p"pT"pn~ 


11J1^ : : 


i i i i i 1 1 i 


i i 1 1 1 1 1 1 



0.1 



111 



R (Rate Measure) 



directly as the measure of the production rate, the definition of a is necessarily changed to the 
theoretical first unit cost in the learning curve when X = 1 and R = 1 . Since the rates for the first 
two buys of our illustrative example are not unity, it is impossible to determine the parameter 
value of the rate term unless there are at least three, and preferably more, data points. 

By combining all available price data when year 3 quotes become available, we can 
derive the parameter values for the Z equation as shown below: 

Ratio Rate Measure Absolute Size Rate Measure 



a = 
b = 
c = 



$100,000 
-0.23445 
-0.321915 



$440,352 

-0.23445 (85% learning curve) 

-0.321915 (80% rate curve) 



The only difference in results is the first unit cost, a. The high value of the first unit 
cost when using the absolute size rate measure is due to the implicit assumption that it is for 
X=1 and R=1, which is outside the relevant production rate range and, therefore, is not a 



11 



meaningful number. 

CHANGING COST STRUCTURE 

The second major issue facing the use of rate adjustment model for weapon systems 
cost estimation is the changing cost structure as a result of changes in production setup. Any 
additional investments in a plant's facilities, whether for capacity expansion or for more efficient 
production methods, alter the cost structure. This change of cost structure does not create 
a significant problem for the X term in Equation (1), since it captures the effect of cumulative 
production experience (a continuous phenomenon). The changing cost structure, however, 
poses a serious question about the suitability of using multi-year cost data for cost models 
involving rate adjustments. The production rate term captures the effect of spreading fixed 
costs over varying numbers of units. During the early stages of production, the amount of 
fixed costs may vary from period to period because of the changing production setup. 
Therefore, the effect of production rate on unit costs may not stabilize until after the production 
setup and its inherent cost structure is stabilized. Trying to derive a rate curve with historical 
data from only the early stages of production is probably unreliable. 

Let us extend the previous example by assuming that the plant capacity is expanded 
in year 3 to accommodate the higher quantity required. The resultant higher fixed costs push 
up the total production curve for any given quantity level from TC1 to TC2, as shown in Figure 
3. TC3 represents the total cost curve if the capacity is further expanded. Figure 4 depicts 
the cost reduction curves under different production rates after the learning curve effect has 
been considered (see Column 3 in Table 1). 



12 



Figure 3 
Total Cost vs Production Rate 



Total Cost 




100 IM 

Quantity (Rate) 



HO SOO 



Figure 4 
Changing Cost Structure & Rate Curve 



Unit Cost 








VOs. 




Erroneou* Rat* Curr* 




*^\^ 








r 


- RC2 




B 


c 


- RCI 



100 150 200 

CJuanUty (Rate) 



2&0 300 



If the government procured 50 units in year 1 under the cost structure labeled TC1 
and RC1, 100 units in year 2 under TC2 and RC2, and 200 units in year 3 under TC3 and 
RC3, the unit costs to the government, after considering the learning curve effect, would be 
Points A, B', and C. Deriving a rate curve using A, B', and C would result in an erroneous 
rate curve, as shown in Figure 4. The slope of the erroneous rate curve is biased by the 
changing cost structure. 

On the other hand, if there is no change in the plant's cost structure, the same cost 
curve (TC1 or RC1) applies to years 1 through 3, and the three data points (A, B, and C in 



13 



Figure 4) would all fall on the same curve (RC1). Therefore, the data would be appropriate for 
estimating the parameter value for the rate term. The same is also true for step-ladder quotes 
for any particular year, which reflect the spreading of fixed costs in a particular year (Points 
A, B, and C) and, therefore, are also appropriate for estimating the parameter values using 
Equation (1). 





Table 2 


The Effect of Changing Cost Structure on Unit Costs 
(a = 100,000, LC = 85%, R = 80%) 


Lot # 


Quantity Total Cost Unit Cost 




A. Same Cost Structure: 


1 

2 
3 


50 $3,191,511 $63,830 
100 3,437,523 34,375 
200 5,601,184 18,671 




B. Changing Cost Structure: 


1 
2 
3 


50 $3,191,511 $63,830 
100 3,837,523 38,375 
200 6,401,184 21,337 



The issue discussed above can be illustrated with a numerical example as shown in 
Table 2. Data for Scenario A are constructed by assuming that there was no change in the 
cost structL ? in the contractor's plant. Data for Scenario B are constructed by adding 
$400,000 and $800,000 of additional fixed costs to year 2 and year 3 total costs respectively. 
Using the three data points under each scenario to derive the parameters for Equation (1) 
results in the following: 



14 



Scenario A Scenario B: 

a = $100,000 $72,227 

b = -0.2344 (85%) -0.1389 (91%) 

c = -0.3219 (80%) -0.3959 (76%) 

It can be seen clearly that analysis of data from Scenario A results in correct 
parameters, while analysis of data from Scenario B distorts all three parameters. What we can 
conclude is that using cost data obtained from a plant which has experienced a changing cost 
structure violates the statistical requirement of drawing samples from a homogeneous 
population. The consequence of sampling from different populations is the distortion of all 
parameters, as shown above. 

CONCLUSIONS 

In this paper, we examined the conceptual underpinning of the production rate effect 
on weapon system costs as well as various production rate measures for rate adjustment 
models. The first conclusion is that the production rate term should be measured as a ratio, 
not as an absolute quantity of lot size or annual/monthly quantity. Expressing the production 
rate as a function of a base rate within the relevant range allows the analyst to estimate the 
learning curve from scanty historical data with more confidence as well as adjust costs for the 
applicable rate effect. It also facilitates the comparison of current step-ladder quotes with the 
historical contract awards. 

There are several other practical considerations that favor the use of a ratio as the 
rate measure. The data base available for learning curve and rate curve determination is 
typically scanty. Using unity as the rate base requires both X and R as the independent 



15 



variables in parameter determination. Having to use two independent variables reduces the 
degrees of freedom and increases the estimating error accordingly. 

The second conclusion is that a stringent condition must be met before an analyst can 
use multi-year cost data to derive parameter values for the widely used rate adjustment model 
(Equation 1). The condition is that the underlying cost structure (variable/fixed cost mix and 
direct/indirect cost mix) must remain the same for all time periods covered by the data. This 
condition is met by step-ladder quotes for various quantities within the same period or by a 
plant that has stabilized its production capacity and setup. Unless this condition is fulfilled, the 
rate adjustment cost model may significantly distort the parameters. We believe that the 
inconclusive findings of prior research regarding production rate impact on weapon systems 
cost can be partially attributed to this problem. 



16 



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Balut, S. J. (1981), "Redistributing Fixed Overhead Costs," Concepts, Vol. 4, No. 2, pp. 
63-72. 

Balut, S. J., T. R. Gulledge, Jr., and N. K. Womer (1989), "A Method of Repricing Aircraft 
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Boger, D. C. and S. S. Liao (1988), Quantity-Split Strategy Under A Competitive 
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Bohn, M. and L A. Kratz (1984), 'The Impact of Production Rate on Unit Costs," Paper 
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Bemis, J. C. (1981), "A Model for Examining the Cost Implications of Production Rate," 
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Cox, L W. and J. S. Gansler (1981), "Evaluating the Impact of Quantity, Rate, and 
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Gulledge, T. R., Jr. and N. K. Womer (1986), The Economics of Made-to-Order 
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Hirsch, W. Z. (1952), "Manufacturing Progress Functions," The Review of Economics and 
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Large, J. P., K. Hoffmayer, and F. Kontrovich (1974), Production Rate and Production 
Cost, R-1609-PA&E, The RAND Corporation, Santa Monica, CA. 

Liao, S. S. (1988), 'The Learning Curve: Wright's Model vs. Crawford's Model," Issues 
in Accounting Education, Vol. 3, No. 2, pp. 302-315. 

Liao, S. S. (1989), "Modifications and Extensions: Applying the Learning Curve 
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Under, K. P., and C. R. Wilbourn (1973), The Effect of Production Rate on Recurring 
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Cost Research Symposium, Airlie, VA, pp. 276-300. 

Smith, L. L. (1976), An Investigation of Changes in Direct Labor Requirements Resulting 
From Changes in Airframe Production Rate, Ph.D. dissertation, University of Oregon, Eugene, 
OR. 



17 



Womer, N. K. (1984), "Estimating Learning Curves From Aggregate Monthly Data," 
Management Science, Vol. 30, pp. 982-992. 



18 



DISTRIBUTION UST 

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19 



Department of Administrative Sciences Library 1 

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Department of Administrative Sciences 
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Naval Postgraduate School 
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